Agricultural Studies 3300
Agricultural Systems Modeling I
Fall 2008
Assignment # 6
Due Friday October 31st , 2008, no later than 8:00 AM
Please hand in your assignment at the START of class
1.
ALBERTA PLAINS TURKEY FARM
The Alberta Plains Turkey Farm is considering buying two different brands of turkey feed and blending them to provide a
good, low-cost diet for its turkeys. Each feed contains, in varying proportions, some or all of the three nutritional
ingredients (protein, vitamin, and iron) essential for fattening turkeys. Each kilogram of brand A contains 10 units of
protein, 8 units of vitamin, and 1 unit of iron. Each kilogram of brand B contains 20 units of protein, 6 units of vitamins,
but no iron. The brand A feed costs the farm 4 cents a kilogram, and the brand B feed costs 6 cents a kilogram. The
owner of the farm would like to use LP to determine the lowest-cost diet that meets the minimum monthly intake
requirement for each nutritional ingredient. The minimum monthly requirement for turkeys is 180 units of protein, 96
units of vitamins and 3 units of Iron.
If we let A denote the number of kilograms of brand A feed purchased and B denote the number of kilograms of brand B
feed purchased, we can proceed to formulate this LP problem as follows:
Minimize cost (in cents) = 4 A + 6 B
2.
Consider the following four LP formulations. Using a graphical approach, determine
(a)
(b)
(c)
(d)
which formulation has more than one optimal solution.
which formulation is unbounded.
which formulation is unfeasible.
which formulation has a unique optimal solution.
Formulation 1
maximize:
subject to :
Formulation 3
10X + 10Y
2X ≤ 10
2X + 4Y ≤ 16
4Y ≤ 8
X ≥ 6
X, Y ≥ 0
Formulation 2
maximize :
subject to :
maximize :
subject to :
3X + 2Y
X + Y ≥ 5
X ≥ 2
2Y ≥ 8
X, Y ≥ 0
Formulation 4
X + Y
X ≤1
2Y ≤ 2
X + 2Y ≤ 2
X, Y ≥ 0
maximize :
subject to :
3X + 3Y
4X + 6Y ≤ 48
4X + 2Y ≤ 12
3Y ≥ 3
2X ≥ 2
X, Y ≥ 0
3.
(FERTILIZER BLENDING) The Sweet Smell Fertilizer Company markets bags of manure labeled “not less
than 60 kilograms dry weight.” The packaged manure is a combination of compost and sewage wastes. To provide
good-quality fertilizer, each bag should contain at least 30 kilograms of compost but no more than 40 kilograms of
sewage. Each kilogram of compost costs Sweet Smell 20 cents and each kilogram of sewage costs 16 cents. Use a
graphical LP method to determine the least cost blend of compost and sewage in each bag.
4.
(AGRICULTURE PLANNING) The seasonal yield of olives in a Piraeus, Greece, vineyard is greatly
influenced by a process of branch pruning. If olive trees are pruned every two weeks, output is increased. The pruning
process, however, requires considerably more labour than permitting the olives to grow on their own and results in a
smaller olive. It also, though, permits olive trees to be spaced closer together. The yield of one barrel of olives by
pruning requires five hours of labour and half a hectare of land. The production of a barrel of olives by the normal
process requires only two labour hours but takes 1 hectare of land. An olive grower has 250 hours of labour available
and a total of 75 hectares for growing. Because of the olive size difference, a barrel of olives produced on pruned trees
sell for $20, whereas a barrel of regular olives has a market price of $30. The grower has determined that because of
uncertain demand, no more than 40 barrels of pruned olives should be produced. Use graphical LP to find
(a)
(b)
(c)
the maximum possible profit.
the best combination of barrels of pruned and regular olives.
the number of hectares that the olive grower should devote to teach growing process.